Bounce inside equilateral
A friction-less board has the shape of an equilateral triangle of side length meter with bouncing walls along the sides. A tiny super bouncy ball is fired from vertex towards the side . The ball bounces off the walls of the board nine times before it hits a vertex for the first time. The bounces are such that the angle of incidence equals the angle of reflection. The distance travelled by the ball in meters is of the form , where is an integer. What is the value of ?
The answer is 31.
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1 solution
Not a rigorous proof, but rather some thoughts. Here is a construction that is helpful.
One may a create such grids of the triangles (all equilateral with the same side length) and The bouncing is similar to passing through a side. All vertices with the same label are equivalent. Then, we only need to start from the lowest A and travel a straight line to the highest B , in the photo. The path would be connecting the vertices of a parallelogram with sides 1 and 5 . The final solution is 3 1 .